We show that, given a set of randomly distributed wireless nodes with density n, when the transmission range rn of wireless nodes satis£es πr n ≥ 3 log n+c(n) n , the localized Delaunay triangulation (LDel) [1] is the same as the Delaunay triangulation with high probability, where c(n) → ∞ as n goes in£nity. Our experiments show that the delivery rates of existing localized routing protocols ar...

Given a set S of points in the plane, a triangulation of S is a partition of the convex hull of S into triangles whose vertices are the points of S. A triangulation of S is said to be Delaunay if no point of S lies inside the triangles’ circumcircles. In this thesis, we study a generalization of these notions to a set S of disjoint segments in the plane. At first, we define a new family of diag...

An intrinsic discrete Laplace-Beltrami operator on simplicial surfaces S proposed in [2] was established via an intrinsic Delaunay tessellation on S. Up to now, this intrinsic Delaunay tessellations can only be computed by an edge flipping algorithm without any provable complexity analysis. In the paper, we show that the intrinsic Delaunay triangulation can be obtained from a duality of geodesi...

TD-Delaunay graphs, where TD stands for triangular distance, is a variation of the classical Delaunay triangulations obtained from a specific convex distance function [5]. In [2] the authors noticed that every triangulation is the TD-Delaunay graph of a set of points in R, and conversely every TD-Delaunay graph is planar. It seems natural to study the generalization of this property in higher d...

When a triangulation of a set of points and edges is required, the constrained Delaunay triangulation is often the preferred choice because of its well-shaped triangles. However, in applications like terrain modeling, it is sometimes necessary to have flexibility to optimize some other aspect of the triangulation, while still having nicely-shaped triangles and including a set of constraints. Hi...

Let P be a polyhedral domain occupying a convex volume. We prove that the size of a graded mesh of P with bounded vertex degree is within a factor O(H P ) of the size of any Delaunay mesh of P with bounded radius-edge ratio. The term HP depends on the geometry of P and it is likely a small constant when the boundaries of P are fine triangular meshes. There are several consequences. First, among...

The problem of computing the exact stretch factor (i.e., the tight bound on the worst case stretch factor) of a Delaunay triangulation has been open for more than three decades. Over the years, a series of upper and lower bounds on the exact stretch factor have been obtained but the gap between them is still large. An alternative approach to solving the problem is to develop techniques for comp...